Sunderland's renaissance under Martin O'Neill continues to gather momentum. A tie that appeared awkward for the Premier League club when the draw was made posed no problems for a Sunderland side playing with the sort of confidence and conviction that was so badly lacking under Steve Bruce. The statistics say it all: O'Neill has overseen five victories in his seven matches in charge; Bruce managed two in 14 games this season and there was a chastening Carling Cup exit at Brighton within that sequence.

Championship opposition was much more easily overcome here as second-half goals from the impressive Sebastian Larsson and the lively James McClean extinguished Peterborough's hopes of causing an upset. There was more than a touch of good fortune about Larsson's free-kick, which was an intended cross that crept inside the far post. But Sunderland thoroughly deserved a lead that was doubled moments later when McClean, who had earlier struck the woodwork, registered his second goal in the space of six days with a well-taken header that effectively ended the contest.

The applause that the Sunderland players left the field to from the sizeable contingent of travelling supporters among a poor crowd of 8,975 was followed by a cheer moments later, when the fourth-round draw handed them a home tie with their bitter rivals Middlesbrough.

"A big roar went up in the corridor, I think it was maybe the last few Sunderland fans that were hanging around or perhaps it was Lee Cattermole," O'Neill said, alluding to the Sunderland captain's former allegiance to Middlesbrough. "It's a great tie and we're at home so let's see if we can make that count."

This was an accomplished and professional performance from a strong Sunderland team – O'Neill made only two changes from his starting line-up in the 4-1 win against Wigan on Tuesday night – and there was never any sense that they were vulnerable.

Darren Ferguson's players came into the game on the back of a five-match unbeaten run but finished the match knowing that they had been totally outclassed and could have no complaints about the result. Simon Mignolet, the Sunderland goalkeeper, never had a save to make.

At the opposite end of the pitch, Joe Lewis, the Peterborough goalkeeper, was much busier as Sunderland seized the initiative from the outset and capitalised on their opponents' profligacy in possession. The absence of a recognised centre-forward – Stéphane Sessègnon was asked to play alone up front because Nicklas Bendtner was ruled out with a knee problem – made little difference as Sunderland dominated central midfield and looked menacing whenever they attacked on the flanks, where McClean's penetration and Larsson's excellent deliveries caused Peterborough problems throughout.

The only surprise was that it took Sunderland until the 48th minute to take the lead. The Peterborough goal had lived a charmed life before half-time –McClean hit the crossbar with a stinging left-footed shot and Lewis made two decent saves to deny the effervescent Craig Gardner – but the home team's good fortune ran out when Larsson's whipped free-kick from wide on the left eluded everyone, including the Peterborough keeper, and bounced before ending up in the far corner.

"A marvellous delivery," O'Neill said."Seb's goal against Blackburn Rovers [a last-minute free-kick which earned a 2-1 win in O'Neill's first game in charge] was a big, big moment for us. And I just think that he's continued. There's a great spirit among the lads and he epitomises that."

McClean also merited special praise from the manager. The 22-year-old was close to signing for Peterborough in the summer until Sunderland gatecrashed the deal and paid Derry £350,000. He produced another promising display here, further vindicating O'Neill's decision to pluck him from the reserves in the manager's first week at the club and, at the same time, raising questions about how on earth Bruce had failed to give the winger a chance.

"He's played three games. You would think he's played 103," O'Neill said.

There was certainly no way back for Peterborough once McClean arrived at the near post to expertly head home Larsson's corner 10 minutes after the Swede had given Sunderland the lead. Sunderland could afford to cruise towards the final whistle thereafter, leaving Ferguson to reflect on a bitterly disappointing afternoon when his two central defenders, Ryan Bennett and Gabriel Zakuani, were the only Peterborough players to emerge with any credit.

"When you play Premier League opposition – and Sunderland are a good team obviously – I think you need nine players playing at their maximum; we had two, so you are always struggling and up against it when that happens," the Peterborough manager said. "It was probably one of our worst performances of the season, and that's never going to be enough to win a game like today."